Question: Isabella built a time travel machine, but she can't control the destination of her trip. Each time she uses the machine she has a $0.25$ probability of traveling to a time before she was born. During the first year of testing, Isabella uses her machine $5$ times. Assuming that each trip is equally likely to travel before Isabella was born, what is the probability that at least one trip will travel before Isabella was born? Round your answer to the nearest hundredth. $P(\text{at least one before she's born})=$
Strategy In this situation it is much easier to calculate the probability of the event we are looking for (at least one trip to a time before Isabella was born) by calculating the probability of its complement (every trip to a time after Isabella was born), and subtracting from $1$. In other words, we can use this strategy: $P(\text{at least one before she's born})=1-P(5\text{ after she's born})$ Calculations $\begin{aligned} &\phantom{=}P(\text{at least one before she's born}) \\\\ &=1-P(5\text{ after she's born}) \\ \\ &=1-(0.75)^{5} \\ \\ &\approx 1-0.2373 \\ \\ &\approx 0.7627\end{aligned}$ Answer $P(\text{at least one before she's born}) \approx 0.76$